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# 3-1 Proofs

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Withina mathematicsa proofa bya contrapositivea essentiallya meansa thata
wea area sayinga thea propositiona toa bea false.a Ana examplea ofa thisa woulda
bea sayinga thata pa ->a qa thena itsa contrapositivea ofa ~qa ->a ~pa toa bea
equivalent.a Wea saya thisa becausea pa (xa )a ->a qa (a xa )a isa infacta truea fora xa thena
~qa (a xa )a ->a ~pa (a xa )a woulda alsoa bea true.a Thisa woulda bea truea becausea ita isa
waya easiera whena provinga contrapositivea statementa toa solvea ~qa (xa )a ->a
~pa (a xa )a thana thata ofa pa (a xa )a ->a qa (a xa ).a
Witha thata beinga saida proofa bya contradictiona woulda bea applieda whena aa
negationa ofa thea theroma statementa ~pa woulda bea waya easiera toa provea
thana thata ofa pa usinga proof.a Ana examplea ofa thisa woulda bea thata ofa sayinga
therea isa noa largesta evena integer.a Nowa usinga k+2.a Wea cana saya thata ka +a 2a
=a (2n)a +a 2a =a 2(n+1.)a Thena saya thata k+2a isa even,a howevera k+2a isa indeeda
largera thana thata ofa justa k.a Thisa woulda bea aa contradictiona becausea thea
statementa clearlya saysa thata ka isa thea largesta evena integer.a Thisa woulda
leada usa backa toa oura originala claima provinga ita toa bea true.